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Math Expert V
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For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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Question Stats: 59% (02:04) correct 41% (01:49) wrong based on 631 sessions

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For integers x and y, $$2^x + 2^y=2^{30}$$. What is the value of $$x + y$$?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64

This is a property of exponents:

$$2^1 + 2^1 = 2^2$$
$$2^2 + 2^2 = 2^3$$ (because $$2^2 * (1 + 1) = 2^2 * 2$$)
Similarly, $$2^3 + 2^3 = 2^4$$
$$2^4 + 2^4 = 2^5$$
etc

So
$$2^{29} + 2^{29} = 2^{30}$$

$$x + y = 29 + 29 = 58$$

Similarly,

$$3^1 + 3^1 + 3^1 = 3^2$$
$$3^2 + 3^2 + 3^2 = 3^3$$
$$3^3 + 3^3 + 3^3 = 3^4$$
and so on...
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Posts: 61
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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$$2^x+2^y\,=\,2^{30}$$
$$x+y\,=\,?$$

$$2^x\,(1+2^{y-x})\,=\,2^{30}$$
$$1+2^{y-x}\,=\,2^{30-x}$$
$$1\,=\,2^{30-x}-2^{y-x}$$

Difference between any two powers of 2 yield 1 if one of the powers is one and the other is zero
i.e. $$2^1\,-\,2^0\,=\,1$$

--> $$30-x\,=\,1\,\,\,$$ and $$\,\,\,y-x\,=\,0$$
$$x\,=\,29\,\,\,$$ and $$\,\,\,x\,=\,y$$
$$x+y\,=\,58$$

##### General Discussion
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

1/2 * (2^30 + 2^30) = 2^30

2^29 + 2^29 = 2^30

The above equation is similar to 2^x + 2^y =2^30.
so x =29 and y=29

Ans:D
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Ans: D

Solution: 2^x + 2^y =2^30
there must be some value of as we know 2^30 only has 2 as its factor so if i take anything common from LHS of the equation the remaining part must be in form of 2 only.
I can take common if and only if x=y; otherwise the remaining part inside the bracket will become odd
for example lets say x= 14 and y = 16 then 2^14+2^16 = 2^14 (1+2^2)= 2^14 * 5 .. and this will happen for all the values of x and y where x is not equal to y
so now=>
the equation becomes 2^x + 2^x =2^30 because x=y
2^x (1+1)= 2^30
2^x * 2 = 2^30
x+1 = 30
x= 29
and y = 29
x+y = 58
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Joined: 14 Mar 2014
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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4
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

IMO: D

Both sides consists of only powers of 2.

Thus in order to that happen x = y must be the case

if x=y then

$$2^x +2^y =2 ^30$$ ==> $$2^x +2^x =2 ^30$$
$$2(2^x) =2 ^30$$
x+1 =30
x=29
Thus x+y = 29+29 =58
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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5
2
2^1 = 2^0 + 2^0
2^2 = 2^1 + 2^1
2^3 = 2^2 + 2^2
.
.
.
2^30 = 2^29 + 2^29

x + y = 58

Option D
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

2^x+2^y=2^30
Now, 2^30=2^29+2^29=2*2^29
or x+y=29+29=58
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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dkumar2012 wrote:
for example lets say x= 14 and y = 16 then 2^14+2^16 = 2^14 (1+2^2)= 2^14 * 5 .. and this will happen for all the values of x and y where x is not equal to y

This is a great solution. A little more detail for those wanting to see why all cases will not work (not just the one above)

2^x + 2^y = 2^x * (1+2^(y-x)) = 2^30

This means that 1+2^(y-x) has to be a multiple of 2 and this can only be achieved if 2^(y-x) = 1 = 2^0.
Math Expert V
Joined: 02 Sep 2009
Posts: 62499
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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3
6
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

This problem is a good candidate for testing small numbers to find patterns. While your instincts may tell you to simply set x + y equal to 30, small numbers will show that you cannot simply set them equal. If you try $$2^x+2^y=2^6$$, for example you can't find values for x + y = 6 that will set the sum equal to 64. The powers of 2 are:

2

4

8

16

32

64

The only pairing that will work is 32 + 32 = 64, meaning that you need $$2^5+2^5=2^6$$, which should make sense: 2 times 2^5 will equal 2^6. So this example should teach you that you need to add two $$2^{29}$$s together to get to $$2^{30}$$. So x and y are each 29, making the sum x + y equal to 58, answer choice D.
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

2^x+2^Y = 2^30

2^x-30 + 2^y-30 = 1 (Divide LHS and RHS by 2^30)

1/2+1/2 = 1 Therefore 2^x-30=1/2 i.e. x-30 = -1 hence X= 29
similarly y-30 = -1 and y = 29

X+Y = 29+29 = 58

Ans D
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64

You are given 2^x+2^y=2^30 ---> in order to understand the question, try to see what values can y take if you start with fixed values for x ---> remember that x and y MUST be integers.

Thus, lets have x =1 ---> $$2^y=2^{30}-2$$ ---> y can not be an integer. Try a bigger value for x.

x = 15 ---> $$2^y=2^{30}-2^{15}$$ ---> $$2^y=2^{15}(2^{15}-1)$$ , keep trying and you will see that there will always be a 1 inside the bracket except when you write the given expression as

$$2^x+2^y=2^{30}$$ --->$$2^x+2^y=2*2^{29}$$ ---> $$2^x+2^y=2^{29}+2^{29}$$ ---> compare both sides of the equation to get x=y=29 and x+y = 58.

D is thus the correct answer.

Hope this helps.
Math Expert V
Joined: 02 Aug 2009
Posts: 8308
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64

Hi,
here we are adding two different numbers which are power to base 2 resulting into another number which has a base of 2..
the logic is .. if we are adding 2^x and 2^y to get 2^z, only way is to get 2 out of the addition and that is possible only when x and y are same..
so $$2^x+2^y=2^{30}$$..where x =y
can be written as $$2^x+2^x=2^{30}$$
$$2*2^x=2^{30}$$..
$$x+1=30 ..x=29$$..
so $$x+y=29+29=58$$
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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1
Here's a better way to solve the problem (suggested by Anthony Ritz - big thank you! ).

Step 1: factor out the given equation and see if there is a pattern
$$2^x(1 + 2^{y-x}) = 2^{30}$$
Hm, interesting. We know that $$2^x$$ is always even, so $$(1 + 2^{y-x})$$ should also be even since $$2^{30}$$ is even.

For $$(1 + 2^{y-x})$$ to be an even integer, $$2^{y-x}$$ needs to be odd. What situation will $$2^{y-x}$$ be an odd number? When $$2^{y-x} = 1$$!
Hence,$$y-x = 0$$ and consequently $$x=y$$.

Step 2: plug in our findings to the equation
$$2^x + 2^y = 2^x + 2^x = 2 * 2^x = 2^{1+x} = 2^{30}$$
The last two parts of our equation tells us that $$1+x = 30$$.
In conclusion, $$x = y = 29$$.

Answer: $$x + y = 58$$.
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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It's actually really easy to spot a pattern here.

2^2 + 2^2 = 8
2^3 = 8
2^4 + 2^4= 16 + 16 = 32
2^5 = 32

So 2^x +2^y = 2^n
to get x+y you just do 2(n-1).
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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I used a simple tester example to see what would happen if I wanted to find a smaller exponential value of 2.

I used the following:

2^4 = 2^x + 2^y --> 8+8 (i.e. 2^3 + 2^3) satisfies this...

sum of x and y has to be one less than exponent we are looking for. Thus 29+29 = 58 and we have our answer.
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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Bunuel wrote:
For integers x and y, $$2^x + 2^y=2^{30}$$. What is the value of $$x + y$$?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

Love this as a 700 level question. Easy if you keep your head on straight. Plug in 2^2 + 2^2= 8 = 2^3.... Plug in 2^3 +2^3= 16 = 2^4....As we can see the pattern shows that 2^29 + 2 ^29 will equal 2^30. Then we can add 29 + 29 and get 58.

Under 30 seconds oooooooooooooooooooooh yea
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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VeritasPrepKarishma wrote:
For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64

This is a property of exponents:

$$2^1 + 2^1 = 2^2$$
$$2^2 + 2^2 = 2^3$$ (because $$2^2 * (1 + 1) = 2^2 * 2$$)
Similarly, $$2^3 + 2^3 = 2^4$$
$$2^4 + 2^4 = 2^5$$
etc

So
$$2^{29} + 2^{29} = 2^{30}$$

$$x + y = 29 + 29 = 58$$

Similarly,

$$3^1 + 3^1 + 3^1 = 3^2$$
$$3^2 + 3^2 + 3^2 = 3^3$$
$$3^3 + 3^3 + 3^3 = 3^4$$
and so on...

Thanks for making things much easier!! Love your blogs as well. Keep up the great work.
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For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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Alternative approach if you don's see the pattern:

$$2^x + 2^y=2^{30}$$, where $$2^x + 2^y$$=$$(2^x +2^y)^2-2*2^{x+y}=2^{30}$$, where $$(2^x + 2^y)^2=2^{60}$$

$$2^{30}*(2^{30}-1)=2^{x+y+1}$$ ,

we can omit -1 since it almost doesn't change the result, so $$2^{60}= 2^{x+y+1}$$, x+y+1=60. x+y = 59 - yes I know that this is not 58, but this is the closest result. Answer (D)
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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Bunuel wrote:
For integers x and y, $$2^x + 2^y=2^{30}$$. What is the value of $$x + y$$?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

$$2^x + 2^y=2^{30}$$
which can be written as, $$2^{29}*({2^{x-29}} + {2^{y-29}})=2^{30}$$

Hence, $${2^{x-29}} + {2^{y-29}} = 2$$

Since x & y are integers, we got $$x-29 = 0$$ & $$y-29 = 0$$

$$x = 29$$, $$y =29$$, $$x+y = 58$$

Thanks,
GyM
_________________ Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?   [#permalink] 08 Jun 2018, 01:06

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